Summary

Area under a curve: The area bounded by $y = f(x)$, the $x$-axis, and vertical lines $x = a$, $x = b$ is $\int_a^b f(x) \, dx$ (if $f(x) \ge 0$). For $f(x) \le 0$, area is $-\int_a^b f(x) \, dx$.

Area between two curves: The area enclosed by $y = f(x)$ and $y = g(x)$ between $x = a$ and $x = b$ is $\int_a^b |f(x) - g(x)| \, dx$, where $f(x) \ge g(x)$ on $[a, b]$.

Area with respect to $y$: For curves $x = f(y)$ and $x = g(y)$ between $y = c$ and $y = d$, area is $\int_c^d |f(y) - g(y)| \, dy$.

Symmetry: Use symmetry to simplify — e.g., area of a circle $x^2 + y^2 = r^2$ is $4 \int_0^r \sqrt{r^2 - x^2} \, dx$. …